Astonishing as it seems, a great many people couldn't care less whether Gordon Brown continues as Prime Minister of Great Britain, or whether David Cameron can yet prove that Lazarus was an Old Etonian. At the last count there were billions out there asking: "Who?"

Elsewhere on the blue-green, pear-shaped (in every sense) planet, they have - as usual - other things on their minds. For tens of millions of Americans, Mr Brown remains the vaguest of presences, at best the one who, for some odd reason, replaced Tony-I-can-remember-his-face. For tens of millions of Russians, Mr Cameron is non-existent, and likely to remain that way.

No surprises there. How are things in Moldova? Can you name the Prime Minister of Norway? A majority of people on this island are none too clear about the political system operating in Edinburgh, far less the clash of ideas in genuinely exotic locales. All politics is parochial until the world - you remember it; big place; abroad - intrudes.

When that happens, only a handful of states matter. This is bitterly disappointing to the patriots who talk, in London, of "punching above our weight", or muse, in Paris, on the afterglow of la gloire. The difference between the handful and the rest is weighed in terms of influence, generally regional but sometimes global. Crudely put, you can't bother them much, but they can bother the hell out of you.

China, as everyone has noticed, has arrived, steadily adding clients and allies to its strategic "string of pearls" through aid, trade, bribery and none-too-subtle pressure. India, its economy ballooning, is the most populous of all, and perhaps the most ambitious. Brazil, after a slow start, is almost ready to take the podium on behalf of South America. For several obvious reasons, the Middle East as yet possesses no clear contender, but sooner or later the vacuum will be filled.

Fewer than 20 years ago, as the Berlin Wall came down, we were telling ourselves that only one superpower remained. The elder Bush was handing out lectures on a new world order, Made in America. Now we live amid emerging regional blocs, in a world in which our parochial politicians remind us daily that we "can't forget China" and "can't overlook India". In Europe, nevertheless, we overlook the obvious.

The landscape defined by the old rivalry between the United States and the USSR has survived, despite all the odds. Some of the details have changed, and corners of the maps have been redrawn. But very soon now the relationship between Russia and America will again shape the future for all the little would-be powers who stand between and around them.

This time everything is democratic and above board, of course, if by democratic you mean an American polity in which the Clinton family is very likely next year to change places, again, with the Bush tribe. You might have imagined that a nation of 300 million souls would have a deeper pool of aspiring talent to draw from, but apparently not.

In Russia, meanwhile, Vladimir Putin also remains wedded to democracy, if by democracy you mean a freely (more or less) elected autocracy. Troubling he may be, but the former KGB man is brilliantly ingenious. His country's constitution says that he cannot stand again for presidency after his second term ends next March. Fine, he replies, I'll exercise my citizen's rights and stand for the Duma instead. Prime Minister with a hand-picked President? Why not?

Putin, with approval ratings of 80%, will encounter little opposition, largely because most of his opponents are already suppressed, censored, exiled, jailed or dead. He will simply move to a slightly smaller office and run Russia as before, his "democratic" credentials intact, his control over the oil and gas billions that have brought increasing prosperity to the electorate unchallenged. Then, come 2012, he will be free to stand again for the presidency.

As I say, brilliant.

Between the two poles, Putin's Russia and Clinton's America, the rest in the northern hemisphere will pick their way, like it or not. The Russian has been making it clear for several years that a subordinate role is not on his agenda. He was bold even before the oil money began to gush. Now his Russia pushes its near neighbours around, boasts of its new weapons systems and lectures America on its failings. That Russia has also begun to take a closer interest in its dealings with Europe. The traditional Slav ambivalence is turning into assertiveness.

A Clinton in the White House should make for an interesting contrast. Clearly, that is one of Hillary's intentions, but the results are likely to disappoint many and surprise some. She may have cast Bill - out of sight but never out of mind - as a fence-mending global ambassador, but this is no "anyone but Bush", stop-the-war Democrat. Dream on.

It is not clear, for one thing, that Hillary even intends to stop either of America's two publicly acknowledged wars. She still refuses to apologise for voting for the Iraq invasion. She is belligerent, to put it mildly, towards Iran, and was the only Democratic contender to vote for a recent resolution casting that country's Revolutionary Guard as terrorists. She appears to have no intention of amending the Afghanistan "mission". Her revived health-care plan, rather than her foreign policy, now seems like the sop to her party's base.

It is possible that Mrs Clinton is only doing what she needs to do, as an often-reviled woman in the febrile, flag-waving climate of American politics, to succeed. It is possible that Mr Putin means only to ensure stability until such time as his country can develop a reliable multi-party system. Such is his usual spiel, much as Hillary likes to hint at "fundamental" differences with George Bush.

Why, then, have notable Republicans begun to speak warmly of the erstwhile devil woman - and why have some even begun to contribute significantly to a campaign war-chest bigger than anything accumulated by Republican contenders? Why do so many Russians seem to exult when Putin honours the Soviet past while proposing a new, proud imperial future? And where does that leave us?

Unless Mrs Clinton has decided that you can fool all the people, all the time, the idea that America will begin to retreat from its international adventures after November 2008 begins to look very optimistic indeed. Putin, meanwhile, seems content to play his full, aggressive part in any rivalries that might then be available. The contest, when it comes, will no longer be unequal: America has been scarred by Iraq and Russia is no longer a basket case.

Europe will have to tread warily. It cannot return to being America's bulwark against Russia. These days Putin possesses an energy weapon that mocks such antique defences. Nor are the EU and Nato likely to find much common cause with Moscow in the foreseeable future. But if Iraq has taught us anything, it is that America's interests are not Europe's interests, and certainly not Britain's interests. Mrs Clinton shows few signs of embracing the novelty.

One of the old, tiny, left-wing sects used to boast a slogan. "Neither Washington nor Moscow", its newspaper insisted. There are worse policies.